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This work is a probabilistic study of the 'primes' of the Cram\'er model. Weprove that there exists a set of integers $\mathcal S$ of density 1 such that\begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\\hbox{prime} \} \ge \frac{1}{\sqrt{2\pi e}\, }, \end{equation} and that for$b>\frac12$, the formula \begin{equation} \mathbb{P} \{S_n\ \text{prime}\, \}\, =\, \frac{ (1+ o( 1) )}{ \sqrt{2\pi B_n } } \int_{m_n-\sqrt{ 2bB_n\logn}}^{m_n+\sqrt{ 2bB_n\log n}} \, e^{-\frac{(t - m_n)^2}{ 2 B_n } }\, {\rmd}\pi(t), \end{equation} in which $m_n=\mathbb{E} S_n,B_n={\rm Var }\,S_n$,holds true for all $n\in \mathcal S$, $n\to \infty$. Further we prove that forany $0<\eta<1$, and all $n$ large enough and $ \zeta_0\le \zeta\le \exp\big\{\frac{c\log n}{\log\log n}\big\}$, letting $S'_n= \sum_{j= 8}^n \xi_j$,\begin{eqnarray*} \mathbb{P}\big\{ S'_n\hbox{\ $\zeta$-quasiprime}\big\} \,\ge\, (1-\eta) \frac{ e^{-\gamma} }{ \log \zeta }, \end{eqnarray*} according toPintz's terminology, where $c>0$ and $\gamma$ is Euler's constant. We also testwhich infinite sequences of primes are ultimately avoided by the 'primes' ofthe Cram\'er model, with probability 1. Moreover we show that the Cram\'ermodel has incidences on the Prime Number Theorem, since it predicts that theerror term is sensitive to subsequences. We obtain sharp results on the lengthand the number of occurrences of intervals $I$ such as for some $z>0$,\begin{equation}\sup_{n\in I} \frac{|S_n-m_n|}{ \sqrt{B_n}}\le z,\end{equation} which are tied with the spectrum of the Sturm-Liouvilleequation.

Critical probabilistic characteristics of the Cram\'er model for primes and arithmetical properties